It may look a bit scary, but it expands to this series of calculations:

The more terms we calculate, the more accurate it becomes (the next term is 25h4/16384, which is getting quite small, and the next is 49h5/65536, then 441h6/1048576)

Comparing

Just for fun, I used the three approximation formulas, and the two exact formulas (but only the first four terms, so it is still just an approximation) to calculate the perimeter for the following values of a and b:

Circle

Lines

a:

10

10

10

10

10

b:

10

5

3

1

0

Approx 1:

62.832

49.673

46.385

44.65

44.429

Approx 2:

62.832

48.442

43.857

40.606

39.834

Approx 3:

62.832

48.442

43.859

40.639

39.984

Series 1:

62.832

48.876

45.174

43.204

42.951

Series 2:

62.832

48.442

43.859

40.623

39.884

Exact*:

20π

40

* Exact:

When a=b, the ellipse is a circle, and the perimeter is 2πa (62.832... in our example).

When b=0 (the shape is really two lines back and forth) the perimeter is 4a (40 in our example).

They all get the perimeter of the circle correct, but only Approx 2 and 3 and Series 2 get close to the value of 40 for the extreme case of b=0.